Skip to main content
SpringerLink
Log in

Fast construction of irreducible polynomials over finite fields

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We present a randomized algorithm that on inputting a finite field K with q elements and a positive integer d outputs a degree d irreducible polynomial in K[x]. The running time is d 1+ɛ(d)×(log q)5+ɛ(q) elementary operations. The function ɛ in this expression is a real positive function belonging to the class o(1), especially, the complexity is quasi-linear in the degree d. Once given such an irreducible polynomial of degree d, we can compute random irreducible polynomials of degree d at the expense of d 1+ɛ(d) × (log q)1+ɛ(q) elementary operations only.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L.M. Adleman and H. W. Lenstra, Jr., Finding irreducible polynomials over finite fields, Proceedings of the 15th Annual ACM Symposium on Theory of Computing, ACM, Boston, MA, 1983, pp. 350–355.

  2. M. Ben-Or, Probabilistic algorithms in finite fields, 22nd Annual Symposium on Foundations of Computer Science 11 (1981), 394–398.

    Article  Google Scholar 

  3. A. Bostan, Ph. Flajolet, B. Salvy and É. Schost, Fast computation of special resultants, Journal of Symbolic Computation 41 (2006), 1–29.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Bostan, L. González-Vega, H. Perdry and É. Schost, From Newton sums to coefficients: complexity issues in characteristic p, Proceedings of MEGA’05, 2005.

  5. J. von zur Gathen and J. Gerhard, Modern Computer Algebra, second edition, Cambridge University Press, 2003.

  6. J. Giraud, Remarque sur une formule de Shimura-Taniyama, Inventiones Mathematicae 5 (1968), 231–236.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. W. Howe, On the group orders of elliptic curves over finite fields, Compositio Mathematica 85 (1993), 229–247.

    MathSciNet  MATH  Google Scholar 

  8. H. Iwaniec, On the problem of Jacobsthal, Demonstratio Mathematica 11 (1978), 225–231.

    MathSciNet  MATH  Google Scholar 

  9. E. Kaltofen and V. Y. Pan, Parallel solution of Toeplitz and Toeplitz-like linear systems over fields of small positive characteristic, in Proceedings of PASCO’94, Lecture Notes Series on Computing 5, World Scientific Publishing Company, Singapore, 1994, pp. 225–233.

    Google Scholar 

  10. K. S. Kedlaya and C. Umans, Fast modular composition in any characteristic, in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2008, pp. 146–155.

    Google Scholar 

  11. H. W. Lenstra, Jr., Factoring integers with elliptic curves, Annals of Mathematics 126 (1987), 649–673.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. W. Lenstra, Jr., Algorithms for finite fields, in Number Theory and Cryptography (Sydney, 1989), London Mathematical Society Lecture Note Series 154, Cambridge University Press, 1990, pp. 76–85.

  13. H. W. Lenstra, Jr., Finding isomorphisms between finite fields, Mathematics of Computation, Vol. 56, 193 (1991), 329–347.

    Article  MathSciNet  Google Scholar 

  14. H. W. Lenstra, Jr., Complex multiplication structure of elliptic curves, Journal of Number Theory 56 (1996), 227–241.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. W. Lenstra, Jr. and B. de Smit, Standard models for finite fields: the definition, http://www.math.leidenuniv.nl/~desmit, 2008, pp. 1–4.

  16. R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Cambridge, MA, 1983.

    MATH  Google Scholar 

  17. Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics 6, Oxford University Press, 2002.

  18. D. Panario and B. Richmond, Analysis of Ben-Or’s polynomial irreducibility test, Random Structures and Algorithms 13 (1998), 439–456.

    Article  MathSciNet  MATH  Google Scholar 

  19. C. H. Papadimitriou, Computational Complexity, Addison Wesley, Cambridge, MA, 1967.

    Google Scholar 

  20. A. Schönhage, Fast parallel computation of characteristic polynomials by Leverrier’s power sum method adapted to fields of finite characteristic, in Automata, Languages and Programming (Lund, 1993), Lecture Notes in Computer Science 700, 1993, Springer, Berlin, pp. 410–417.

    Chapter  Google Scholar 

  21. J.-P. Serre, Complex multiplication, in Algebraic Number Theory (J. W. S. Cassels and A. Fröhlich eds.), Academic Press, New York, 1967.

    Google Scholar 

  22. V. Shoup, Fast construction of irreducible polynomials over finite fields, in Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms (Austin, TX, 1993), ACM, New York, 1993, pp. 484–492.

    Google Scholar 

  23. J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, Springer-Verlag, Berlin, 1986; expanded 2nd edition, 2009.

    Book  MATH  Google Scholar 

  24. C. Umans, Fast polynomial factorization and modular composition in small characteristic, in Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 1986, pp. 350–355.

  25. J. Vélu, Isogénies entre courbes elliptiques, Comptes Rendus de l’Académie des Sciences, Série I 273 (1971), 238–241.

    MATH  Google Scholar 

  26. W. C. Waterhouse, Abelian varieties over finite fields, Annales Scientifiques de l’École Normale Supérieure, Série 4 2 (1969), 521–560.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA Bordeaux Sud-Ouest & Université de Toulouse, Université de Toulouse le Mirail, Département de Mathématiques et Informatique, 5 allées Antonio Machado, F-31058, Toulouse cédex 9, France

    Jean-Marc Couveignes

  2. La Roche Marguerite, DGA, F-35174, Bruz Cedex, France

    Reynald Lercier

  3. Institut de Recherche Mathématique de Rennes, Université de Rennes 1, Campus de Beaulieu, F-35042, Rennes, France

    Reynald Lercier

Authors
  1. Jean-Marc Couveignes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reynald Lercier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Marc Couveignes.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Couveignes, JM., Lercier, R. Fast construction of irreducible polynomials over finite fields. Isr. J. Math. 194, 77–105 (2013). https://doi.org/10.1007/s11856-012-0070-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-012-0070-8

Keywords

Search

Navigation